Том 215, № 3 (2016)

We prove that the Dirichlet and Poincar´e problems for degenerate multidimensional hyperbolic equations with Gellerstedt operator in a cylindrical domain are uniquely solvable and establish a criterion of uniqueness for the regular solutions of these problems.

We establish asymptotic representations for one class of solutions of a second-order differential equation of the Emden–Fowler type with rapidly varying nonlinearity.

We introduce and study a continuous function I which is called a digit inversor for the Q₃-representation of the fractional part of a real number. This representation is determined by a probability vector.(q₀; q₁; q₂) with positive coordinates, generalizes the classical ternary representation, and coincides with this representation for q₀ = q₁ = q₂ = 1/3: The values of this function are obtained from the Q₃-representation of the argument by the following change of digits: 0 by 2; 1 by 1; and 2 by 0: The differential, integral, and fractal properties of the inversor are described. We prove that I is a singular function for q₀ ≠ q₂.

We consider a homogeneous system of linear singularly perturbed differential equations of order m with matrix at higher derivatives that becomes singular as the small parameter approaches zero. By using the Newton diagrams, we study the structure of the general solution of the analyzed system and the possibility of construction of its asymptotics in the case where the corresponding characteristic polynomial of the matrix has multiple finite and infinite elementary divisors. The obtained results generalize the results obtained for similar systems of equations of the first and second orders.

We establish conditions for the existence of almost periodic and periodic solutions of almost periodic difference equations with discrete argument in a metric space without using the -classes of these equations.

We study the asymptotic behavior of solutions for a certain class of third-order nonlinear nonautonomous differential equations.